## Online Course Discussion Forum

### Hints for Winter Camp Blue Practice Questions

Now that the camp is officially over, I would like to request a few hints on select problems. These are problems that I have difficulty getting started with and would like someone to point me in the right direction. Thanks in advance.

Number Theory P16, P17

Combinatorics P19 - P24

And by the way, Happy New Year!

Charles :D

Number Theory P17: Start by showing that between two consecutive powers of $5$, there are always $2$ or $3$ powers of $2$.

Consider now the intervals $[5^n, 5^{n+1}]$. Some of them will contain two powers of $2$, and some three powers of $2$. Let $x$ be the number of intervals with two powers of $2$, and $y$ the number of intervals with $3$ powers of $2$. Use the fact that $2^{2013} < 5^{867} < 2^{2014}$ to set up a system of equations using $x$ and $y$. Note the answer is precisely $y$.

Combinatorics 19:

Note an outcome like $H[\cdots]HHHHH$ is twice as likely to occur as an outcome like $TH[\cdots]HHHHH$, where the tosses in $[\dots]$ are the same in both .

Let $p$ be the probability of getting the five $H$s before two $T$s with the first flip being $H$, and $q$ the probability of getting the five $H$s before two $T$s with the first flip being $T$. Find (non equivalent) expressions for $q$ in terms of $p$, and $p$ in terms of $q$, then use them to find what $p$ and $q$ are. The answer is then $p + q$.

Combinatorics 20:

If the first three tiles you have drawn are $A$, $B$, and $C$, it must be true that $A=B$, $A=C$, or $B=C$, otherwise the game ends without emptying the bag. Let $p_n$ be the probability of emptying the bag when there are exactly $n$ pairs left to grab from the bag. Use the three cases mentioned above to find relationships between the $p_i$s. Clearly $p_1 = 1$, and $p_6$ is the answer we are looking for.

Combinatorics 21:

Clearly the smallest possible height is $94 \times 4$. If some $4$s are swapped to $10$s, or $19$s, the height can increase by $6$, or $15$, respectively. Say you swap $m$ $4$s to $10$s, and $n$ $4$s to $19$s, then the height increases by $6m + 15n = 3(2m + 5n)$. Thus, you will have as many possible heights as there are numbers of the form $2m + 5n$ for non-negative integers $m$ and $n$ with $n + m \leq 94$.

Combinatorics 23:

Consider the function $$f(x) = \lfloor2x\rfloor + \lfloor4x\rfloor + \lfloor6x\rfloor.$$ Show there is a value of $m$ such that, given $x$, $f(m) = f(x) + 6$. So, it $n = f(x)$ for some $x$, then also $n + 6$ can be written in this form. How can you use this to find all numbers that work?

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